Question: Simplify and expand the following expression: $ \dfrac{2}{p + 1}- \dfrac{4}{4p + 16}- \dfrac{p}{p^2 + 5p + 4} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the second term: $ \dfrac{4}{4p + 16} = \dfrac{4}{4(p + 4)}$ We can factor the quadratic in the third term: $ \dfrac{p}{p^2 + 5p + 4} = \dfrac{p}{(p + 1)(p + 4)}$ Now we have: $ \dfrac{2}{p + 1}- \dfrac{4}{4(p + 4)}- \dfrac{p}{(p + 1)(p + 4)} $ The least common multiple of the denominators is: $ (p + 1)(p + 4)$ In order to get the first term over $(p + 1)(p + 4)$ , multiply by $\dfrac{4(p + 4)}{4(p + 4)}$ $ \dfrac{2}{p + 1} \times \dfrac{4(p + 4)}{4(p + 4)} = \dfrac{8(p + 4)}{(p + 1)(p + 4)} $ In order to get the second term over $(p + 1)(p + 4)$ , multiply by $\dfrac{p + 1}{p + 1}$ $ \dfrac{4}{4(p + 4)} \times \dfrac{p + 1}{p + 1} = \dfrac{4(p + 1)}{(p + 1)(p + 4)} $ In order to get the third term over $(p + 1)(p + 4)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{p}{(p + 1)(p + 4)} \times \dfrac{4}{4} = \dfrac{4p}{(p + 1)(p + 4)} $ Now we have: $ \dfrac{8(p + 4)}{(p + 1)(p + 4)} - \dfrac{4(p + 1)}{(p + 1)(p + 4)} - \dfrac{4p}{(p + 1)(p + 4)} $ $ = \dfrac{ 8(p + 4) - 4(p + 1) - 4p} {(p + 1)(p + 4)} $ Expand: $ = \dfrac{8p + 32 - 4p - 4 - 4p}{4p^2 + 20p + 16} $ $ = \dfrac{28}{4p^2 + 20p + 16}$ Simplify: $ = \dfrac{7}{p^2 + 5p + 4}$